Using a generalisation of optical theorem to \(d=D-1\)  non-compact transverse spatial directions, see [1]}, [2]}, that relies on partial wave expansion in terms of the relevant spherical harmonics (Gegenbauer polynomials, related to characters of \(SU(2)\)  [3]}) and Bessel functions one has (for massless probes \(k=\omega \) )
\(\sigma _\ell (\omega ) = {2^{d-2}\pi ^{{d\over 2}-1} \over 3 \omega ^5} (\ell +{d\over 2}-1)\Gamma ({d\over 2}-1)\left(^{\ell +d-3}_{\quad \ell \quad }\right) (1-|S_\ell |^2)\) 
